Alternative Render engines can I use instead of Cycles for AMD Radeon GPUs?
AMD GPUs have a long history of issues with blender cycles, unfortunately the support for those cards is not 100% due to issues with the drivers.

You can try other rendering engines like the ones listed on the Resources for Blender Some of them have addons that integrate the render engine directly to blender. Some of them you can use for free, others are commercial products. 

External Render engines:


Luxrender  is a physically based and unbiased rendering engine. Supports Blender integration.
Yafaray is a free open-source raytracing engine which supports both biased an unbiased rendering methods. Supports Blender integration.
TheBounty is a free montecarlo raytracing engine. (fork of Yafaray)
Mitsuba is a research-oriented rendering system in the style of PBRT.
Appleseed is a modern, open source, physically-based renderer designed for production rendering.
Jot is a program that lets designers, artists and animators directly annotate 3D computer graphics models (and animation) for stylized (NPR) rendering [in real time]. 

A blender exporter and more up to date info can be found here.
Nox is a free stand-alone, physically-accurate state-of-an-art renderer. Windows only(open-source since 2014 but seems original authors abandoned).
Aqsis is a cross-platform 3D rendering solution adhering to the RenderMan standard.
POV-ray is a high-quality, Free Software tool for creating stunning three-dimensional graphics. Supports Blender integration.
PRMan Another cross platform integration of Pixar's RendermanProServer 20.0+ (Free for non-commercial use).
V-Ray for Blender An add-on that packages a .blend file and converts it into a format that V-Ray standalone knows how to render.
B-Maxwell
The B-Maxwell add-on is a basic exporter from Blender to Maxwell Render (Windows and Linux). "B-Maxwell add-on offers seamless integration with Maxwell Render and Blender"
Octane-Render Commercial plugin for blender available. GPU accelerated Unbiased rendering engine.